2015-01-15

### Introduction

Instruments true to the baroque period are generally made as copies of preserved instrument from that period. It is assumed that the historic instrument maker has succeeded to make an almost perfect construction. But is that really so? Is it possible to improve the construction of a historical instrument type? In that case will the new constructed instrument still be true to the period? I venture to say it will and I have made an attempt to improve the baroque treble recorder. In the following I will explain the findings I have made concerning historical recorders and the aim of my new bore design.

- First a simplified description of the resonant system of the recorder i.e. what is going on in the flute tube.
- Then a simplified description of the tone generation mechanism i.e. what happen when the air jet from the wind way hits the labium edge.

It is important to understand, that here we have two separate phenomena which are partly independent and partly dependent of each other. They cooperate but can also partly counteract.

**The resonant system**

A flute tone will arise in a tube when the air inside it is put into oscillation of an energy impulse. The oscillation occurs when the energy caught in the tube swings between two different states. As with all oscillating systems. An easy way to explain example is an oscillating pendulum where the energy swings between potential energy and kinetic energy. The potential energy is maximal at the endpoints and the kinetic energy at the passage of the lowest point. In between there are both potential and kinetic energy in varying quantity. The total energy is constant along the whole path.

In a flute, the oscillating air also contains kinetic and potential energy. The oscillation can be described in this way: The air flows out in both ends of the flute. An under- pressure will be created in the middle of the flute. It will stop the outflow of air and change it to an inflow. This inflow will build up kinetic energy and continue until the under pressure changes to an over pressure which finally stops the inflow and change that to an outflow etc. At the lowest tone of the flute there is one point at the middle of the pipe with solely potential energy (node) and solely kinetic energy (bellies) at the ends. The length of the flute will in that case be about a half wavelength of the produced tone. It is possible with more than one nod along the pipe. The only condition that must be fulfilled for resonance is that there must be bellies at the ends. With two nods the resonant frequency will be twice, with three nods three times the fundamental etc. An ideal flute will thus have an infinite number or resonant frequencies which are natural number multiples of the fundamental i.e. constitute a harmonic series.

**The tone generation**

How the sound generation starts when air is blown through the flute is so complicated so I will leave that out. The energy supply to maintain an existing oscillation comes from the air jet through the wind channel. When the oscillating air swings outwards at the labium the air jet will bend out and pas on the outside of the flute. Nothing will happen. In a moment a half cycle later the oscillating air swings into the flute and bends the air jet into it and the kinetic energy of the jet adds to the oscillating energy in the right phase to keep it up. The energy supply comes thus in short pulses with the timing controlled of the existing oscillation.

Periodic phenomenon of any kind can always be split up into a number of sinusoid phenomena of different frequency, amplitude and phase. This was discovered by the French mathematician Fourier (1772-1830). He showed that the frequencies of the sinusoids in such a split up were distributed in a harmonic series. From that we can understand that the oscillation controlled of the lowest used resonant frequency of the flute will concentrate the energy from the jet to that frequency and to its natural number multiple frequencies i.e. its harmonics.

With an ideal flute there will be a perfect match between the flute resonant frequencies and the frequency content of the air jet and a strong trumpet like ton will be produced, see fig.1.

##### Fig. 1: Ganassi recorder

This is not always the case. For the baroque recorder, the condition for the lowest tone is as in fig. 2:

##### Fig. 2: Baroque recorder

The resonances are stretched out so the 4th resonance is almost a tone higher than the 1st harmonic. The resonances act as filters for the energy in the harmonics and will thus influence the timbre of the tone (compare the timbre control on an amplifier). The lowest tone on a baroque recorder thus has a timbre based on almost the fundamental alone. Fig. 3:

##### Fig. 3

*For the other tones, the 1st and 2nd resonance can be adjusted to an octave interval through the correct placement and size of the finger holes. This is required, as the same finger hole is to be used for both octaves.*

### The resonance system once again

In a real flute there are a number of conditions which influence the resonance frequencies. We call them: *End correction, Window correction, Hole correction *and* Bore effect.*

##### Fig. 4

*The end correction* is rather small and almost frequency independent and effect only the lowest tone of the flute, so we leave that out in this discussion.

*The window correction* comes from the fact that the window opening has an area smaller than the area of the flute bore. In passing this obstacle, the oscillating air will lose some potential energy, so the swing from potential- to kinetic energy will not be fully completed inside the flute. This can be looked upon as the oscillation continuing a small distance outside the flute. This virtual lengthening is frequency dependent and the largest at low frequencies. Fig. 4 shows to the left, in a bar chart, the window correction for the lowest tone and the three first harmonics for a normal treble recorder.

*The hole correction* is present at all resonances where open finger holes are used. This correction comes also from the difference in area of the finger hole and the bore, but is more complicated because a part of the swinging energy will pass through the finger hole and a part will continue further down the flute bore. We can imagine that more of the energy in low frequencies will find its way out through the hole, and more of that in the higher will pass. In fact, an open finger hole shortens the acoustic length of the flute more for a low tone than a high. The frequency dependence is thus the opposite to that of the window correction. Furthermore it is possible to get the same lowest resonance with a small hole closer to the window or a larger hole further down, but the displacement of the higher resonances will be larger with a small hole. The flute maker is to congratulate, as this fact makes it possible to choose a place for the hole where the right resonance for the low frequency is achieved and the right hole correction for compensating the window correction for the second resonance. Fig.4 shows to the right two alternative hole positions for the same low resonance and the influence on the higher resonances.

*The bore effect* comes from local variations in potential and kinetic energy caused by variations in the bore cross section area. A shrinking of the area around the node point in the middle of the flute will reduce the volume for the air and cause the potential energy to increase or decrease faster. This will increase the resonance frequency. A similar shrinking at the end of the flute will slow down the in and out passage of the air and reduce the resonance frequency. Same type of area change will thus cause opposite change in frequency, depending on the dominating type of energy at the place. We then understand that a shrinking in the middle will increase the first resonance and decrease the second which has a belly there, whereas a shrinking at the end will decrease both resonance frequencies. Fig.5 shows the influence of a single short shrinking at any place along a flute bore on the first thee resonances.

##### Fig. 5

These relations were probable to some extent known from experience by flute makers already during the renaissance. J W S Rayleigh gave it a mathematic formulation in 1896 and it could at that time for instance be used for calculation of the influence of a single deformation on an organ pipe. To calculate the influence from a complete irregular conic flute bore the problem must be split up in a large number of point for point calculations and summed up resonance for resonance. This will demand a computer. I have developed such a computer program and made calculations on some historical recorders.

### Calculation result

The bore dimensions come from measurements made by Fred Morgan on instruments from well known historic recorder makers on tree instruments in the recorder collection of Frans Brüggen and one in Musikhistorisk Museum Köpenhamn.

Fig. 6-9 show the results of these calculations. The diagrams show the bore effect in groups for the four first resonances, for the full length of the flute and for the lengths corresponding to each finger hole.

##### Fig. 6:

##### Fig. 7

##### Fig. 8

##### Fig. 9

### Discussion

It is possible to see both similarities and differences between the designs.

There is much of similarity for the tone T0. The first resonance has been flattened with 60-70 cent compared to straight cylindrical bore. (For a cylindrical bore all the bars will be zero.) This explains why a baroque recorder is shorter than a Ganassi flute in the same pitch. The second resonance is 30-50 cent sharp to the first, and the third and fourth 70-85 cent. With the window correction added the fourth resonance will be about 100 cent higher than the first. This creates the necessary condition for playing the high tones d#, e and f on the baroque treble recorder.

For the tones T2-T9 the sharpening of the second resonance relative the first is rather similar for all designs. This similarity comes from the need to tune the finger holes to function in both octaves and to produce reasonably good semitones at cross fingering, which is very sensitive to hole size. The displacement of the resonances by the window correction and the bore effects must thus be compensated to an octave by finger holes, which also will produce correct semitones at cross fingering. T11-T14 are not used for playing the second octave but the tuning of the second resonance will influence the timbre. There were thus good reasons for the historic flute makers to find bores which fulfilled these requirements, otherwise the recorders would not have functioned at all.

Study of the third and fourth resonances for T2-T14 exposes dissimilarities. There is no system at all either from tone to tone or between different makers. This is however not astonishing as the historic makers had no means to find out this connections. Differences in timber are possible to hear but to connect that to a suitable change of the bore is practically impossible.

### Hypothesis

As the window correction and the hole correction both are proportional to the frequency it would probably be favourable with an equally proportional bore effect. (Fig. 4) It should give the summed effects an uniform pattern of resonances for all the tones and probably result in most equal timber all over the instrument range.

I was pondering for many years on how that could be achieved. Luckily, one day I was struck by the relationship between the equations showed in fig. 5 and the in technical science well-known fourier transform and got the idea from my engineering experience that a bore based on an exponential function probably could give that. So it did. But a bore solely based on an exponential function couldn’t give the large displacement between the first and fourth resonances needed for the function of the baroque fingering of the high tones. Fortunately, some small adjustments of the bore, close to the bottom end, could cure that without destroying the uniformity. Fig. 10 shows the result of this new bore:

##### Fig. 10

### Result

I have built a number of treble recorders in high and low pitch (type BA440 and BA415) with this bore, and they have met appreciation for a very stable low range and an equal timber and fast attack over the whole range.

Are those recorders still baroque instruments? I claim they are, which I think will be clear from fig. 11, which shows my bore and the four others. (The diagram shows bore diameter -10mm. The span is thus 10-20mm). The small steps in the curves come from rounding to 0,1mm, the larger bends in the BA415 curve are compensations for the bore cross section increases due to the finger holes. The whole bore can be described as a sum of mathematical functions.

##### Fig. 11

© Ragnar Arvidsson 2015